# On the oscillation of higher order dynamic equations

Journal of Advanced Research (2013) 4, 201–204

Cairo University

SHORT COMMUNICATION

On the oscillation of higher order dynamic equations
Said R. Grace

*

Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza 12221, Egypt
Received 4 March 2012; revised 2 April 2012; accepted 23 April 2012
Available online 2 July 2012

KEYWORDS

Abstract

Oscillation;
Higher Order;
Dynamic equations



aðtÞðx

We present some new criteria for the oscillation of even order dynamic equation

DnÀ1

ðtÞÞa

D

þ qðtÞðxðtÞÞa ¼ 0;

on time scale T, where a is the ratio of positive odd integers a and q is a real valued positive rdcontinuous functions deﬁned on T.

(ii) a, q: T ﬁ R+ = (0, 1) is a real-valued rd-continuous
functions, aD(t) P 0 for t 2 [t0, 1)T and

Introduction
This paper is concerned with the oscillatory behavior of all
solutions of the even order dynamic equation


nÀ1

aðtÞðxD ðtÞÞa

D

þ qðtÞðxðtÞÞa ¼ 0;

ð1:1Þ

on an arbitrary time-scale T ˝ R with Sup T = 1 and n P 2 is

an even integer.
We shall assume that:
(i) a P 1 is the ratio of positive odd integers,

* Tel.: +20 2 35876998.
Peer review under responsibility of Cairo University.

Production and hosting by Elsevier

Z

1

aÀ1=a ðsÞDs ¼ 1:

ð1:2Þ

We recall that a solution x of Eq. (1.1) is said to be nonoscillatory if there exists a t0 2 T.
Such that x(t)x(r(t)) > 0 for all t 2 [t0, 1)T; otherwise, it is
said to be oscillatory. Eq. (1.1) is said to be oscillatory if all its
solutions are oscillatory.
The study of dynamic equations on time-scales which goes
back to its founder Hilger [1] as an area of mathematics that
has received a lot of attention. It has been created in order
to unify the study of differential and difference equations.
Recently, there has been an increasing interest in studying
the oscillatory behavior of ﬁrst and second order dynamic
equations on time-scales, see [2–7]. With respect to dynamic
equations on time scales it is fairly new topic and for general
basic ideas and background, we refer to [8,9].
It appears that very little is known regarding the oscillation
of higher order dynamic equations [10–15] and our purpose
here to establish some new criteria for the oscillation criteria
for such equations. The obtained results are new even for the
special cases when T = R and T = Z.

http://dx.doi.org/10.1016/j.jare.2012.04.003

202

S.R. Grace
Theorem 2.2. Let conditions (i), (ii) and (1.2) hold and

Main results

Z

We shall employ the following well-known lemma.


Z
rðsÞ ðaðsÞÞÀ1

1

t0

Lemma 2.1. [6, Corollary 1]. Assume that n 2 N, s, t 2 T and
f 2 Crd(T, R). Then
Z

t

s

Z

t

ÁÁÁ
gn þ1

¼ ðÀ1Þn

Z

Z

Z
t!1

t

ð2:1Þ

Â
Ã
gðsÞqðsÞ À aðsÞgD ðsÞhÀa
nÀ1 ðs; t0 Þ Ds ¼ 1;

ð2:4Þ

t1

t

hn ðs; rðgÞÞfðgÞDg:

ð2:3Þ

If there exists a positive non-decreasing delta-differentiable
function g such that for every t1 2 [t0, 1)T.
lim sup

gn

1=a
qðuÞDu
Ds ¼ 1:

s

t

fðg1 ÞDg1 Dg2 Á Á Á Dgnþ1

1

then Eq. (1.1) is oscillatory.

s

We shall employ the Kiguradze’s following well-known
lemma.
Theorem 2.1. (Kiguarde’s Lemma [8, Theorem 5]),Let
n 2 N; f 2 Cnrd ðT; RÞ and sup T = 1. Suppose that f is either
n
positive or negative and fD is not identically zero and is either
nonnegative or nonpositive on [t0, 1)T for some t0 2 T. Then,
there
exist
t1 2 [t0, 1)Tm 2 [0, n)Z
such
that
n
ðÀ1ÞnÀm fðtÞfD ðtÞ P 0 holds for all t 2 [t0, 1)T with
j

(I) f ðtÞf D ðtÞ > 0 holds for all t 2 [t0, 1)T and all
j 2 [0, m)z,
j
(II) ðÀ1Þmþj f ðtÞf D ðtÞ > 0 holds for all t 2 [t0, 1)T and all
j 2 [m, n)z

Proof. Let x(t) be a nonoscillatory solution of Eq. (1.1) on
[t0, 1)T. It sufﬁces to discuss the case x is eventually positive
(as – x also solves (1.1) if x does), say x(t) > 0 for t P t1 P t0.
Now, we see that

D
nÀ1
aðtÞðxD ðtÞÞa 6 0 for t P t1 :
nÀ1

It is easy to see that xD ðtÞ > 0 for t P t1 for otherwise,
and by using condition (1.2) we obtain a contradiction to the
fact that x(t) > 0 for t P t1. Now, aD(t) P 0 for t 2 [t0, 1)T.
We have

D
 nÀ1
D
 nÀ1 a
nÀ1
aðtÞðxD ðtÞÞa ¼ aD ðtÞ xD ðtÞ þ ar ðtÞ ðxD ðtÞÞa
6 0:

The following result will be used to prove the next corollary.
Lemma 2.2. [12, Lemma 2.8].Let supT = 1 and
f 2 Cnrd ðT; RÞðn P 2Þ. Moreover, suppose that Kiguradze’s theon
rem holds with m 2 [1, n)N and fD 6 0 on T. Then there exists a
sufﬁciently large t1 2 T such that
m

fD ðtÞ P hmÀ1 ðt; t1 ÞfD ðtÞ for all t 2 ½t1 ; 1ÞT :

This implies
nÀ1

ððxD ðtÞÞa ÞD 6 0 for t P t1 :
nÀ1

Next, we let y ¼ xD
rem 1.90] we see that

ð2:2Þ

Z

on [t1, 1)T. From Ref. [9, Theo-

1

ðy þ hlyD ÞaÀ1 dh P ayD

Z

1

The proof of the following corollary follows by an integration
of (2.2).

0 P ðya ÞD ¼ ayD

Corollary 2.1. Assume that the conditions of Lemma 2.2 hold.
Then

Thus we have yD ¼ xD 6 0 on [t1, 1)T and from Theorem 2.1, there exists an integer m 2 {1, 3, . . . , n À 1}. Such that
(I) and (II) hold on [t1, 1)T Clearly xD(t) > 0 for t P t1 and
hence, there exists a constant c > 0 such that

m

fðtÞ P hm ðt; t1 ÞfD ðtÞ for all t 2 ½t1 ; 1ÞT :
Next, we need the following lemma see [16].
Lemma 2.3. If X and Y are nonnegative and k > 1, then

0

yaÀ1 dh

0

¼ ayaÀ1 yD :
n

xðtÞ P c

ð2:5Þ

for t P t1 :

First, we claim that m = n À 1. To this end, we assume that

Xk À kXYkÀ1 þ ðk À 1ÞYk P 0;

xD ðtÞ < 0

where equality holds if and only if X = Y,

Integrating Eq. (1.1) from t P t1 to u P t, letting u ﬁ 1 we
have

It will be convenient to employ the Taylor monomials (see
[9, Section 1.6]) fhn ðt; sÞ1
n¼0 g which are deﬁned recursively by:
Z t
hn ðs; sÞDs; t; s 2 T and n P 1;
hnþ1 ðt; sÞ ¼

nÀ2

nÀ3

and xD ðtÞ > 0


Z
nÀ1
xD ðtÞ P c ðaðtÞÞÀ1

for t P t1 :

1=a

1

for t P t1 :

qðsÞDs
t

s

where it follows that h1(t, s) = t À s but simple formulas in
general do not hold for n P 2.
Now we present the following oscillation results for Eq.
(1.1).

Integrating (2.6) from to and letting ﬁ 1 we get
nÀ2

0 < ÀxD ðtÞ 6 À

Z
t

1


Z
ðaðuÞÞÀ1
u

1=a

1

qðsÞDs

Du:

ð2:6Þ

On the oscillation of higher order dynamic equations

203

Integrating this inequality from t and using condition (2.3)
after Lemma 2.1 we arrive at the desired contradiction. It
follows from Lemma 2.2 with m = n À 1 that
DnÀ1

D

x ðtÞ P hnÀ2 ðt; t1 Þx

ðtÞ for t P t1 ;

ð2:7Þ

and by applying Corollary 2.1 with m = n À 1 instead of Lemma 2.2, we get
nÀ1

xðtÞ P hnÀ1 ðt; t1 ÞxD ðtÞ

 D
  D 
g
g
x
wD 6 Àgq þ r wr À a r
wr :
g
g
x
Using Lemma 2.2 with m = n À 1 in (2.15), we ﬁnd
!
 D
 
nÀ1
g
g
xD
r
wr
hnÀ2
w 6 Àgq þ r w À a r
g
g
x
D

nÀ1

DnÀ1 a

aðx Þ
xa

on ½t1 ; 1ÞT :

ð2:9Þ

 g D

g

nÀ1

axD
x

ð2:10Þ

Now set
ð2:11Þ

a

X ¼ ðaðaÀ1=a ÞghnÀ2 Þaþ1


wD ðtÞ 6 ÀgðtÞqðtÞ þ aðtÞgD ðtÞðhnÀ1 ðt; t1 ÞÞÀa ;

¼
t P t1 :

Integrating this inequality from t2 > t1 to t P t2, we have
t

½gðsÞqðsÞ À aðsÞgD ðsÞhÀa
nÀ1 ðs; t1 ÞDs:

ð2:12Þ

t2

Taking upper limit of both sides of the inequality (2.12) as
t ﬁ 1 and using (2.4) we obtain a contradiction to the fact
that w(t) > 0 on [t1, 1)T. This completes the proof. h
Next, we establish the following result.

t!1

Â hnÀ2 ðs; t0 ÞgðsÞ

ð2:13Þ

x þ lhxD

ÃaÀ1

a
ða þ 1Þaþ1



1
ghnÀ2

a

ðgD Þaþ1

on ½t2 ; 1ÞT :

ð2:18Þ

Integrating (2.18) from t2 to t, we get
Z t"

gðsÞqðsÞ À

aðsÞ



1

a

ða þ 1Þaþ1 gðsÞhnÀ2 ðs;t1 Þ

#
ðgD ðsÞÞaþ1 Ds:

Taking upper limit of both sides of (2.19) as t ﬁ 1 and
using (2.13), we obtain a contradiction to the fact that
w(t) > 0 for t P t1 This completes the proof. h
Finally, we present the following interesting result.

Proceeding as in the proof of Theorem 2.1, we obtain
m = n À 1 and (2.7) and (2.8). Deﬁne w as in (2.9) and obtain
(2.10). Now from Ref. [3, Theorem 1.90],
ðxa ÞD ¼ ðaxD Þ

;

ð2:19Þ

Proof. Let x be a nonoscillatory solution of Eq. (1.1), say
x(t) > 0 for t P t1 P t0.

a !a
!aþ1

and therefore, we ﬁnd

t2

then Eq. (1.1) is oscillatory.

Z

a1=a
aghnÀ2

P 0;

aðsÞðgD ðsÞÞaþ1
aþ1

Ds ¼ 1:

and Y

in Lemma 2.3 with k ¼ aþ1
> 1 to conclude that
a
!

 D
g
1
g
a
ðgD Þaþ1
r 1þ1=a
ðw Þ
À r wr þ
a
aþ1
1þ1=a
hnÀ2
g
ga hanÀ2
ða þ 1Þ
ðgr Þ

wðtÞ 6 wðt2 Þ À

ða þ 1Þ
!Àa #

t1

 r
w
g

a
a
ðgD Þa
aþ1

wD 6 Àgq þ

Theorem 2.3. Let conditions (i), (ii) (1.2) and (2.3) hold. If
there exists a a positive non-decreasing delta-differentiable
function g such that for every t1 2 [t0, 1)T
1

ð2:17Þ

for t P t2 P t1 ;

Using (2.8) in (2.11), we get

Z t"
lim sup
gðsÞqðsÞ À

on ½t2 ; 1ÞT ;

!
 D
g
ðaÞÀ1=a g
r
ðhnÀ2 Þðwr Þ1þ1=a
w 6 Àgq þ r w À a
g
ðgr Þ1þ1=a

ðaðxD Þa Þr þ a ðaðxD Þa ÞD
" x
#
D a
a D
DnÀ1 a r g x À gðx Þ
;
¼ Àgq þ ðaðx Þ Þ
xa ðxr Þa
!a
DnÀ1
D x
6 Àgq þ ag
:
x

Z

 1=a
 r 1=a
w
w
P
g
g

D

nÀ1

xa

wðtÞ 6 wðt2 Þ À

¼

and thus,

Then on [t1, 1)T, we have
wD ¼

ð2:16Þ

where hnÀ2 = hnÀ2(t, t1). Now we see that

Now, we let
w :¼ g

for t

P t2 P t1 ;

ð2:8Þ

for t P t1 :

ð2:15Þ

dh P axD

0

Z

Theorem 2.4. Let conditions (i), (ii), (1.2) and (2.3) hold. If
there exists a a positive, delta-differentiable function g such that
for every t1 2 [t0, 1)T
Z t"
lim sup
t!1

1

xaÀ1 dh ¼ axaÀ1 xD :

t1

#
 
1
ðaðsÞÞr ðgD ðsÞÞ2
gðsÞqðsÞ À
Ds
4a gðsÞðhnÀ1 ðs; t1 ÞÞaÀ1 hnÀ2 ðs; t1 Þ

¼ 1;

0

ð2:20Þ

ð2:14Þ

Using (2.14) in (2.10), we have

then Eq. (1.1) is oscillatory.

204

S.R. Grace

Proof. Let x be a nonoscillatoy solution of Eq. (1.1), say
x(t) > 0 for t P t1 P t0
Proceeding as in the proof of Theorem 2.3, we obtain (2.17)
which cam be rewritten as
!
 D
g
ðaÞÀ1=a g
r
w 6 Àgq þ r w À a
ðhnÀ2 Þ
g
ðgr Þ1þ1=a

2. We may also employ other types of the time-scales [8,9] e.g.,
T = hZ with h > 0; qN o ; q > 1, T ¼ N 20 , etc. The detail are

References

D

r 1=aÀ1

Â ðw Þ

r 2

ðw Þ

on ½t2 ; 1ÞT :

ð2:21Þ

Now, using Corollary 2.1 with m = n À 1 we have
x
nÀ1
xD

P hnÀ1 ;

implies on [t2, 1)T that (2.22)
!1Àa
DnÀ1
1=aÀ1
1=aÀ1 1=aÀ1 x
w
¼a
g
x

aÀ1
x
¼ a1=aÀ1 g1=aÀ1 DnÀ1
P a1=aÀ1 g1=aÀ1 haÀ1
nÀ1 :
x

ð2:22Þ

Using (2.22) in (2.21) we have on [t2, 1)T that
wD 6 Àgq þ

 D
g
gr

À
¼ Àgq À
þ

 aÀ1 r 
gðh Þ h
wr À a ðaÞnÀ1r ðgr ÞnÀ2
ðwr Þ2 :
2

r
ahÀ1
nÀ2 ðhnÀ1 Þ

aÀ1

Á1=2

g

À

ððaÞr Þ1=2 gr
r

ðaÞ

ðgD Þ2

4aðhrnÀ1 Þ

6 Àgq þ

aÀ1

hnÀ2


À1Á
4a

!2

ððaÞr Þ1=2 gD
aÀ1

2ðahnÀ2 ðhrnÀ1 Þ

gÞ1=2

:
ðaÞr ðgD Þ2

gðhrnÀ1 Þ

aÀ2

hnÀ2


:
ð2:23Þ

Integrating this inequality from t2 to t, taking upper limit of
the resulting inequality as t ﬁ 1, and applying condition (2.20)
we obtain a contradiction to the fact that w(t) > 0 for t P t1.
This completes the proof.

h

Remarks
1. The results of this paper are presented in a form which is
essentially new and of high degree of generality. Also, we
can easily formulate the above conditions which are new sufﬁcient for the oscillation of Eq. (1.1) on different time-scales
e.g., T = R and T = Z. The details are left to the reader.

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